First, we derive the lower bound on \(C^{\mathrm {(M)}}\) .
For the effective channel in (REF ), it is known that when \(M\rightarrow \infty \) , the double-reflection channel dominates the others and the direct S-R channel is negligible [1]}.Note that in [1]}, only the double-reflection link is considered.
Thus, to obtain the lower bound on \(C^{\mathrm {(M)}}\) , we propose to design the cooperative IRS passive beamforming to maximize the double-reflection channel power gain only, and at the same time, make the two single-reflection channels phase-aligned with the double-reflection channel.
Note that it will be shown by simulations that this cooperative IRS passive beamforming design achieves favorable rate performance even when \(M\)  is small, and moreover does not compromise the capacity scaling order.
To this end, we first decompose the inter-IRS channel between I\(_1\)  and I\(_2\) , \(G_{\rm I_{1},\rm I_{2}}\) , as follows.
\(\begin{aligned}G_{\rm I_{1},\rm I_{2}}\!\!=\!\!\underbrace{\sqrt{{g}_{\rm I_{1},\rm I_{2}}}a_{\rm r}\!\!\left(\theta ^{\rm r}_{\rm I_{1},\rm I_{2}},\!\vartheta ^{\rm r}_{\rm I_{1},\rm I_{2}},\!M_2\right)}_{t_{1}}\underbrace{\sqrt{{g}_{\rm I_{1},\rm I_{2}}}a_{\rm t}^H\!\!\left(\theta ^{\rm t}_{\rm I_{1},\rm I_{2}},\!\vartheta ^{\rm t}_{\rm I_{1},\rm I_{2}},\!M_1\right)}_{t^H_{2}}.\end{aligned}\) 
